THE ALTERNATIVE DUNFORD-PETTIS PROPERTY IN SUBSPACES OF OPERATOR IDEALS

Title & Authors
THE ALTERNATIVE DUNFORD-PETTIS PROPERTY IN SUBSPACES OF OPERATOR IDEALS

Abstract
For several Banach spaces X and Y and operator ideal $\small{\cal{U}}$, if $\small{\cal{U}}$(X, Y) denotes the component of operator ideal $\small{\cal{U}}$; according to Freedman's definitions, it is shown that a necessary and sufficient condition for a closed subspace $\small{\cal{M}}$ of $\small{\cal{U}}$(X, Y) to have the alternative Dunford-Pettis property is that all evaluation operators $\small{\phi_x\;:\;\cal{M}\;{\rightarrow}\;Y}$ and $\small{\psi_{y^*}\;:\;\cal{M}\;{\rightarrow}\;X^*}$ are DP1 operators, where $\small{\phi_x(T)\;=\;Tx}$ and $\small{\psi_{y^*}(T)\;=\;T^*y^*}$ for $\small{x\;{\in}\;X}$, $\small{y^*\;{\in}\;Y^*}$ and $\small{T\;{\in}\;\cal{M}}$.
Keywords
Dunford-Pettis property;Schauder decomposition;compact operator;operator ideal;
Language
English
Cited by
1.
Strongly alternative Dunford–Pettis subspaces of operator ideals, Ukrainian Mathematical Journal, 2013, 65, 4, 649
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